since there exist integersxxx and yyy such that x⁢p+y⁢q=1xpyq1xp\!+\!yq=1 and consequently 1∈(p,q)1pq1\in(p,\,q). Therefore, the principal ideals (p)p(p) and (q)q(q) of ??\mathcal{O} have no common prime ideal factors. Because there are infinitely many rational prime numbers, also the corresponding principal ideals have infinitely many different prime ideal factors.